Cool ways of defining functions.

[TylerH]

$$f(x)=x^3$$
or
$$f:\Re \rightarrow \Re$$
$$r \rightarrow r^3$$

What are some other ways to define functions? Exotic and extraneous as possible.

 

[JJacquelin]

I suppose that you now a cool way to define the function f(x) = ln(x)

 

[TylerH]

Yeah, that was my first example. :)

 

[HallsofIvy]

$$f(x)=x^3$$
or
$$f:\Re \rightarrow \Re$$
$$r \rightarrow r^3$$

What are some other ways to define functions? Exotic and extraneous as possible.

Actually, $f(x)= x^3$ does NOT define a function because it does not specify the domain. You might well assume that the real numbers is intended but why not the complex numbers.

As for the second form, it says too much. You don't have to specify that the range is the set of real numbers because if the domain is the real numbers and the "formula" is $x^3$, the range must be the set of real numbers.

If you want an exotic way of defining functions, how about this:
Bessel's function, of order 0 and of the "first kind" is defined as
"The function satisfying Bessel's equation of order 0,
$$x^2\frac{d^2y}{dx^2}+ x\frac{dy}{dx}+ x^2y= 0$$
and the initial conditions y(0)= 1, y'(0)= 0."

Or the "Lambert W function" which is defined as
"The inverse function to $f(x)= xe^x$".

 

[CellCoree]

There are countless ways to define functions, some of which may seem unconventional or even strange. One example could be using a piecewise function, where the function's definition changes based on the input value. For instance, we could define a function f(x) as follows:

f(x) = \begin{cases} 2x & \text{if } x \text{ is rational} \\ \frac{1}{x} & \text{if } x \text{ is irrational} \end{cases}

This function would have a different definition depending on whether the input is a rational or irrational number. Another way to define a function could be using a recursive formula, where the function's value at a certain point is dependent on its previous values. For example, we could define a function g(x) as:

g(x) = \begin{cases} 1 & \text{if } x = 0 \\ g(x-1) + 2 & \text{if } x > 0 \end{cases}

This function would have a different definition for every input greater than 0, with each definition being dependent on the previous one. Additionally, we could define a function using a power series, where the function is expressed as an infinite sum of terms. For instance, we could define a function h(x) as:

h(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{n!} x^n

This function would have a different definition for every value of x, with each definition being a specific term in the power series. As you can see, there are countless ways to define functions, each with its own unique properties and applications.